Problem: A standard six-sided die is rolled, and  $P$ is the product of the five numbers that are visible. What is the largest number that is certain to divide $P$?
Explanation: Since $6! = 720 = 2^4 \cdot 3^2 \cdot 5$, the prime factors of $P$ can consist of at most 2's, 3's, and 5's.  The least possible number of 2's is two, which occurs when 4 is not visible. The least possible number of 3's is one, which occurs when either 3 or 6 is not visible, and the least number of 5's is zero, when 5 is not visible.  Thus $P$ must be divisible by $2^2\cdot3 =
\boxed{12}$, but not necessarily by any larger number.